Limit

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What is the expansion of sin(x) in terms of x?

x + x^2 + x^3 +...
x^2 - x^4/3! + x^6/5! -...
x - x^3/3! + x^5/5! -... (correct)
1 - x^2/2! + x^4/4! -...

What is the limit of f(x) as x approaches a, if f(x) is an algebraic function?

It is a logarithmic limit
It is a trigonometric limit
It is a exponential limit
It is a algebraic limit (correct)

What is the expansion of cos(x) in terms of x?

1 + x + x^2/2! + x^3/3! +...
1 - x^2/2! + x^4/4! - x^6/6! +... (correct)
x^2 - x^4/3! + x^6/5! -...
x - x^3/3! + x^5/5! -...

What is the expansion of tan(x) in terms of x?

x + x^3/3! + x^5/5! +... (A) Signup and view all the answers

What is the expansion of e^x in terms of x?

1 + x + x^2/2! + x^3/3! +... (B) Signup and view all the answers

What is the value of lim x → 0 (sin 2x / x)?

2 (A) Signup and view all the answers

If f(x) = 2 sin(x) and g(x) = 3 cos(x), then what is the value of lim x → 0 (f(x) / g(x))?

2/3 (D) Signup and view all the answers

What is the derivative of the function f(x) = tan(2x) using the chain rule?

2 sec^2(2x) (A) Signup and view all the answers

What is the value of lim x → 1 (G(x) - G(1)) / (x - 1) if G(x) = 25 - x^2?

1/24 (C) Signup and view all the answers

What is the value of lim x → a (g(x) f(a) - g(a) f(x)) / (x - a) if f(a) = 2, f'(a) = 1, g(a) = 1, and g'(a) = 2?

3 (A) Signup and view all the answers

What is the value of lim(x→∞) ((x+1)^2)/(x^2+2x)?

1 (A) Signup and view all the answers

If a, b, c, and d are positive, then what is the value of lim(x→∞) ((a+bx)/(c+dx))?

e^(d/b) (B) Signup and view all the answers

What is the value of lim(x→∞) (1+x)^(1/x)?

e (A) Signup and view all the answers

What is the value of lim(x→0) (sin(x)/x)?

1 (A) Signup and view all the answers

What is the value of lim(x→∞) (x^2+x-1)/(x^2+2x)?

1 (C) Signup and view all the answers

What is the value of lim h → 0 (1 - cos(2h)) / sinh?

1/2 (B) Signup and view all the answers

What is the value of lim x → 0 (x^2 sin(3x)) / sin(5x)?)

0 (C) Signup and view all the answers

What is the value of lim x → 0 (tan(x) - sin(x)) / x^2?)

1/6 (B) Signup and view all the answers

What is the value of lim x → 0 (sin(2x) / sin(5x))?)

2/5 (A) Signup and view all the answers

What is the value of lim x → 1 (f(x) - f(1)) / (x - 1) if f(x) = 2 sin(x)?)

2 (C) Signup and view all the answers

What is the value of lim x → 0 (sin(x) / x^2)?)

1/3 (C) Signup and view all the answers

What is the value of lim x → 0 (cos(2x) - 1) / sin(x)?)

-2 (A) Signup and view all the answers

What is the value of lim x → 0 (x^3 sin(3x)) / sin(5x)?)

0 (B) Signup and view all the answers

What is the value of lim x → 0 (sin(3x) / sin(5x))?)

3/5 (B) Signup and view all the answers

What is the value of lim x → 0 (tan(2x) - sin(2x)) / x^2?)

1/6 (A) Signup and view all the answers

What is the Taylor series expansion of the function log(1 + x) around x = 0?

x - x^2/2 + x^3/3 -... (A) Signup and view all the answers

What is the condition for the function f(x) = (1 + 1/x)^x to be continuous at x = 0?

The function is not continuous at x = 0. (C) Signup and view all the answers